For normal distributed random variables it is easy to find $F_{|X|}$ given $F_X$ because of the symmetry of the distribution. I'd like to find $F_{|X|}$ for any distribution of $X$.
$F_{|X|} = Pr[|X| \leq x] = Pr[-x\leq X \leq x] = \int_{-x}^x f_X(t) dt = [F_X(t)]_{-x}^x = F_X(x)-F_X(-x)$
I'm not sure if this proof is valid for any distribution of $X$.
Your proof is valid, though for completeness you should also note that $F_{|X|}(x)=0$ for $x\le0$.